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Properties

Label 2-37-37.27-c5-0-3
Degree $2$
Conductor $37$
Sign $-0.905 - 0.423i$
Analytic cond. $5.93420$
Root an. cond. $2.43602$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.33 − 5.39i)2-s + (14.0 + 24.4i)3-s + (42.1 + 73.0i)4-s + (−41.0 + 23.6i)5-s − 304. i·6-s + (−18.9 − 32.7i)7-s − 564. i·8-s + (−276. + 478. i)9-s + 510.·10-s − 364.·11-s + (−1.18e3 + 2.05e3i)12-s + (604. − 349. i)13-s + 407. i·14-s + (−1.15e3 − 667. i)15-s + (−1.69e3 + 2.93e3i)16-s + (−1.23e3 − 710. i)17-s + ⋯
L(s)  = 1  + (−1.65 − 0.953i)2-s + (0.904 + 1.56i)3-s + (1.31 + 2.28i)4-s + (−0.733 + 0.423i)5-s − 3.44i·6-s + (−0.145 − 0.252i)7-s − 3.11i·8-s + (−1.13 + 1.96i)9-s + 1.61·10-s − 0.908·11-s + (−2.38 + 4.12i)12-s + (0.992 − 0.572i)13-s + 0.556i·14-s + (−1.32 − 0.766i)15-s + (−1.65 + 2.86i)16-s + (−1.03 − 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.905 - 0.423i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ -0.905 - 0.423i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0871284 + 0.392198i\)
\(L(\frac12)\) \(\approx\) \(0.0871284 + 0.392198i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (5.14e3 - 6.54e3i)T \)
good2 \( 1 + (9.33 + 5.39i)T + (16 + 27.7i)T^{2} \)
3 \( 1 + (-14.0 - 24.4i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (41.0 - 23.6i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (18.9 + 32.7i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 364.T + 1.61e5T^{2} \)
13 \( 1 + (-604. + 349. i)T + (1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (1.23e3 + 710. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (759. - 438. i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 - 2.21e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.80e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.56e3iT - 2.86e7T^{2} \)
41 \( 1 + (580. + 1.00e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 - 1.80e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.60e4T + 2.29e8T^{2} \)
53 \( 1 + (1.41e4 - 2.45e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.91e3 + 1.10e3i)T + (3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (9.93e3 - 5.73e3i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.10e3 - 1.91e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-3.13e4 - 5.42e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 1.26e4T + 2.07e9T^{2} \)
79 \( 1 + (5.23e4 - 3.02e4i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (1.65e4 - 2.87e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-1.42e4 - 8.20e3i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 896. iT - 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.70392652146380175015158278943, −15.54554250611022993265037938457, −13.36730358083482887171196562482, −11.32671101468923701288694819317, −10.67601218853061101728288350295, −9.720410486250526310023397959935, −8.597432989107395951648194638000, −7.70149868485029315087275284858, −3.87588411185292419319825465197, −2.77410326015184600841870927674, 0.33645989392063489219780682851, 1.99774954703931831465618986230, 6.29407524472412594061339241208, 7.30986265164386826489710917768, 8.492843560732432786227852541381, 8.776682800288655848812290410109, 10.87899890194137250112516177900, 12.48365737847341455807471835157, 13.89691515062138796609126007125, 15.22196859304263084418023819672

Graph of the $Z$-function along the critical line